Untangling a Planar Graph

  • Authors:

  • Xavier Goaoc;Jan Kratochvíl;Yoshio Okamoto;Chan-Su Shin;Andreas Spillner;Alexander Wolff

  • Affiliations:

  • LORIA–INRIA Grand Est, Nancy, France;Charles University, Department of Applied Mathematics and Institute of Theoretical Computer Science, Prague, Czech Republic;Tokyo Institute of Technology, Graduate School of Information Science and Engineering, Tokyo, Japan;Hankuk University of Foreign Studies, School of Electronics and Information Engineering, Yongin, Korea;University of East Anglia, School of Computing Sciences, Norwich, UK;Technische Universiteit Eindhoven, Faculteit Wiskunde en Informatica, Eindhoven, The Netherlands

  • Venue:
  • Discrete & Computational Geometry
  • Year:
  • 2009

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Abstract

A straight-line drawing δ of a planar graph G need not be plane but can be made so by untangling it, that is, by moving some of the vertices of G. Let shift(G,δ) denote the minimum number of vertices that need to be moved to untangle δ. We show that shift(G,δ) is NP-hard to compute and to approximate. Our hardness results extend to a version of 1BendPointSetEmbeddability, a well-known graph-drawing problem. Further we define fix(G,δ)=n−shift(G,δ) to be the maximum number of vertices of a planar n-vertex graph G that can be fixed when untangling δ. We give an algorithm that fixes at least $\sqrt{((\log n)-1)/\log\log n}$vertices when untangling a drawing of an n-vertex graph G. If G is outerplanar, the same algorithm fixes at least $\sqrt{n/2}$vertices. On the other hand, we construct, for arbitrarily large n, an n-vertex planar graph G and a drawing δ G of G with $\ensuremath {\mathrm {fix}}(G,\delta_{G})\leq \sqrt{n-2}+1$and an n-vertex outerplanar graph H and a drawing δ H of H with $\ensuremath {\mathrm {fix}}(H,\delta_{H})\leq2\sqrt{n-1}+1$. Thus our algorithm is asymptotically worst-case optimal for outerplanar graphs.